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Is there an established or proven relationship concerning how much data can fit in a given volume of space / spacetime?

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The limit you want is the https://en.m.wikipedia.org/wiki/Bekenstein_bound bekenstien limit. This limit states that the amount of information a given volume can contain is proportional to the surface area of that volume. Specifically it is 1 bit of information for every 4 plank areas.

Note that because the limit is a surface area proportional to $r^2$ and the volume is proportional to $r^3$ where r is the radius, large volumes must be less information dense then is possible for some smaller volumes. With the limit of information of the universe identified with the "edge" of the observable universe.

Another way to understand this is if you try to push information smaller then this then you will condense it into a black hole with the event horizon being the correct area.

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  • $\begingroup$ Additional details scholarpedia.org/article/Bekenstein_bound $\endgroup$ Jul 9 at 15:04
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    $\begingroup$ Interesting. Maybe it's another question, but can you elaborate on how can two smaller volumes can have greater information capacity than the sum of their volumes? I'd think you could just partition any volume of space into increasingly smaller volumes and see the information capacity grow without bound, based solely on arbitrary definitions of the volumes. $\endgroup$ Jul 9 at 16:54
  • $\begingroup$ if to many small information dense volumes come together into to small a large volume then you get a blackhole whose size would be large enough that is you can only have so many small dense regions in a large volume. $\endgroup$ Jul 9 at 17:45
  • $\begingroup$ also to quickly note this is closely related to the holographic principle if you are interested in further reading. $\endgroup$ Jul 9 at 17:55

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